Interval estimation of stress-strength reliability parameter for exponential-inverted exponential model: Frequentist and Bayesian approaches

Document Type : Original Scientific Paper

Author

Department of Statistics, Faculty of Science, Fasa University, Fasa, Iran

Abstract

This paper introduces the problem of interval estimation for stress strength reliability parameter P(X<Y)‎, ‎where random variables X and Y stand for stress and strength‎, ‎respectively‎. ‎In most of the research papers‎, ‎the authors assumed that X and Y come from the same family of distribution‎. ‎By taking into account some situations arise‎, ‎in this paper we assume that X and Y follow exponential and inverted exponential distributions‎, ‎respectively‎. ‎Our goal is to construct a confidence interval for reliability parameter in this model by using some (approximately) exact and strong methods such as bootstrap‎, ‎generalized and highest posterior distribution approaches‎. ‎Also‎, ‎we compare these methods by means of the expected length and coverage probability criteria‎. ‎Finally‎, ‎a real data set is given and we apply the above methods of estimation on it to inference about the parameter of interest.

Keywords

Main Subjects

References

Ahmad, K.E., Fakhry, M.E. and Jaheen, Z.F. (1995). Bayes estimation of P(Y ≤ X) in the geometric case. Microelectronic Reliability, 35(5):817–820.
Andrews, L. (1985). Special Functions for Engineers and Applied Mathematicians. Macmillan Publishing Company, A Division of Macmillan, Inc., New York.
Babayi, S., Khorram, E. and Tondro, F. (2014). Inference of P[X<Y] for generalized logistic distribution. Statistics, 48:862–871.
Bai, X., Shi, Y., Liu, Y. and Liu, B. (2019). Reliability inference of stress–strength model for the truncated proportional hazard rate distribution under progressively type-II censored samples. Applied Mathematical Modelling, 65:377–389.
Baklizi, A. and Quader El-Masri, A.E. (2004). Shrinkage estimation of P(X < Y) in the exponential case with common location parameter. Metrika, 59:163–171.
Berger, J.O. (1985). Statistical Decision Theory and Bayesian Analysis. New York: Wiley.
Birnbaum, Z.W. (1956). On a Use of Mann-Whitney Statistics. Proc. Third Berkeley Symp. in Math. Statist. Probab., I:13–17.
Chen, M.H. and Shao, Q.M. (1999). Monte Carlo estimation of Bayesian credible and HPD intervals. Journal of Computational and Graphical Statistics, 8(1):69–92.
Constantine, K., Carson, M. and Tse, S. (1990). Confidence interval estimation of P(Y < X) in the gamma case. Journal of Statistical Computation and Simulation, 19:225–244.
Constantine, K. and Karson, M. (1986). The estimation of P(Y < X) in gamma case. Journal of Statistical Computation and Simulation, 15:365–388.
Downtown, F. (1973). On the estimation of Pr(Y < X) in the normal case. Technometrics, 15:551–558.
Efron, B. (1988). Logistic regression, survival analysis, and the Kaplan-Meier curve. Journal of the American Statistical Association, 83:414–425.
Efron, B. and Tibshirani, R. (1994). An Introduction to the Bootstrap. Chapman & Hall/CRC Press.
Ferguson, T.S. (1996). A Course in Large Sample Theory. New York: Chapman and Hall.
Genç, A. (2013). Estimation of P(X>Y) with Topp-Leone distribution. Journal of Statistical Computation and Simulation, 83:326–339.
Govindarajulu, Z. (1967). Two sided confidence limits for P(X > Y) based on normal samples of X and Y. Sankhya, 29:35–40.
Hall, P. (1988). Theoretical comparison of bootstrap confidence intervals. The Annals of Statistics, 16(3):927–953.
Ismail, R., Jeyaratnam, S.S. and Panchapakesan, S. (1986). Estimation of Pr[X >Y] for gamma distributions. Journal of Statistical Computation and Simulation, 26:253–267.
Jovanović, M. (2017). Estimation of P{X < Y} for geometric-exponential model based on complete and censored samples. Communications in Statistics - Simulation and Computation, 46:3050–3066.
Kakade, C.S., Shirke, D.T. and Kundu, D. (2008). Inference for P(Y < X) in exponentiated Gumbel distribution. Journal of Statistics and Applications, 3:121–133.
Khalifeh, A., Mahmoudi, E. and Chaturvedi, A. (2020). Sequential fixed-accuracy confidence intervals for the stress–strength reliability parameter for the exponential distribution: two-stage sampling procedure. Computational Statistics. doi:10.1007/s00180-020-00957-5.
Knight, K. (2000). Mathematical Statistics. Chapman & Hall/CRC Press.
Kotz, S., Lumelskii, Y. and Pensky, M. (2003). The Stress-Strength Model and its Generalizations: Theory and Applications. World Scientific Publishing Co. Pte. Ltd.
Krishnamoorthy, K., Mukherjee, S. and Guo, H. (2007). Inference on reliability in two-parameter exponential stress-strength model. Metrika, 65:261–273.
Kundu, D. and Gupta, R. (2005). Estimation of P[Y < X] for generalized exponential distribution. Metrika, 61:291–308.
Kundu, D. and Gupta, R. (2006). Estimation of P[Y < X] for Weibull distributions. IEEE Transactions on Reliability, 61:270–280.
Mahmoudi, E., Khalifeh, A. and Nekoukhou, V. (2019). Minimum risk sequential point estimation of the stress-strength reliability parameter for exponential distribution. Sequential Analysis, 38(3):279–300.
Maiti, S.S. (1995). Estimation of P(X ≤ Y) in the geometric case. Journal of Indian Statistical Association, 33:87–91.
Makkar, P., Srivastava, P., Singh, R. and Upadhyay, S. (2014). Bayesian survival analysis of head and neck cancer data using lognormal model. Communications in Statistics - Theory and Methods, 43:392–407.
McCool, J. (1991). Inference on P[Y < X] in the Weibull case. Communications in Statistics Simulation and Computation, 20:129–148.
Nadarajah, S. (2004). Reliability for Laplace distributions. Mathematical Problems in Engineering, 2:169–183.
Nadarajah, S. (2005a). Reliability for some bivariate beta distributions. Mathematical Problems in Engineering, 1:101–111.
Nadarajah, S. (2005b). Reliability for some bivariate gamma distributions. Mathematical Problems in Engineering, 2:151–163.
Nadarajah, S. and Kotz, S. (2006). Reliability for some bivariate exponential distributions. Mathematical Problems in Engineering, 2006:1–14.
Obradović, M., Jovanović, M., Milošević, B. and Jevremović, V. (2015). Estimation of P{X ≤ Y} for geometric-Poisson model. Hacettepe Journal of Mathematics and Statistics, 44:949–964.
Owen, D.B., Craswell, K.J. and Handson, D.L. (1964). Non-parametric upper confidence bounds for P(Y < X) and confidence limits for P(Y < X) when X and Y are normal. Journal of American Statistical Association, 59:906–924.
Pak, A., Khoolenjani, N.B. and Jafari, A.A. (2014). Inference on P(Y < X) in bivariate Rayleigh distribution. Communications in Statistics - Theory and Methods, 43(22):4881–4892.
Rezaei, S., Tahmasbi, R. and Mahmoodi, M. (2010). Estimation of P[Y<X] for the generalized Pareto distribution. Journal of Statistical Planning and Inference, 140:480–494.
Sathe, Y.S. and Dixit, U.J. (2001). Estimation of P(X <Y) in the negative binomial distribution. Journal of Statistical Planning and Inference, 93:83–92.
Sharma, V. (2017). Bayesian analysis of head and neck cancer data using generalized inverse Lindley stress-strength reliability model. Communications in Statistics - Theory and Methods, 47(5):1155–1180.
Singh, S., Singh, U. and Kumar, D. (2013). Bayes estimators of the reliability function and parameter of inverted exponential distribution using informative and non-informative priors. Journal of Statistical Computation and Simulation, 83(12):2258–2269.
Tong, H. (1977). On the Estimation of P(Y<X) for exponential families. IEEE Transactions on Reliability, 26:54–56.
Watson, G.N. (1944). A Treatise on the Theory of Bessel Functions. Cambridge University Press.
Weerahandi, S. (1993). Generalized confidence intervals. Journal of the American Statistical Association, 88:899–905.
Weerahandi, S. (1995). Exact Statistical Methods for Data Analysis. Springer Verlag, New York.
Woodward, W.A. and Kelley, G.D. (1977). Minimum variance unbiased estimation of P(X < Y) in the normal case. Technometrics, 19:551–558.

Files

History

  • Receive Date: 26 January 2020
  • Revise Date: 15 March 2020
  • Accept Date: 27 March 2020

Share

How to cite

Statistics